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Syftet med studien är att undersöka hur lågstadielärare arbetar med problemlösning utifrån frågeställningarna vilka strategier lyfter lärare fram i problemlösningsundervisningen och vilken roll tar läraren i undervisningssituationer av problemlösning. Fär att samla in epiri till studien har tre lärare deltagit och deras före, under och efter problemlösningsundervisningen undersöktes. Lärarnas planering av undervisningen gav information om tankar och utgångspunkter i problemlösningen som sedan utfördes och som då observerades med hjälp av två observationscheman utifrån det teoretiska ramverket, bestående av Pólyas fyra faser (1945) och Smith och Steins fem undervisningspraktiker (2014). Observationerna kompletterades med intervjuer av observerade lärare. Insamlad data bearbetades och sammanställdes till ett resultat. Analyserades gjordes av resultatet med det teoretiska ramverket som filter. Studiens resultat visar att introduktionen av ett problem är viktigt för elevers förståelse för det. Lärarens förberedelse och förutsättande av lösningsmetoder och strategier innan lektionen bidrar till att de lyckades bättre med att tydliggöra den matematiska idén i diskussionsdelen av problemlösningens. Samtidigt visar resultatet att lärare lägger olika mycket vikt på att uppmärksamma utvärderingar och reflektion i problemlösningen. Övergripande visar studien att lärarna tar en stor roll i att hjälpa och kontrollera elevernas process i lösandet av problem. Det framkommer även att lärarnas aktiva urval av elevlösningar skiljer sig mellan lärarna.

Made available in DSpace on 2017-07-21T20:56:27Z (GMT). No. of bitstreams: 1 Fabricio Marom Moura.pdf: 5154244 bytes, checksum: 47f0642926df01b449301ba504e49fe7 (MD5) Previous issue date: 2015-08-25 / The main focus of this work is to present the solution of problems supported by softwares as a pedagogical alternative in order to improve the teaching and the learning of mathematics content. In order to do this, the present work connects the aim of the National Curricular Parameters (PCN's), the technique described by Pólya in his book "How to solve it", along the possibility of utilization of programs such as: GeoGebra, Graphmatica, Spreadsheets and Maxima. It is known that many mathematics educators have shown interest for this subject of research. However, despite of all enthusiasm, few of them can put it into practice in the teaching-learning process. The solution of problems still exists as a complementary activity and is sometimes confused with solving exercises where the answer is not analyzed and the values found have no meaning for the student. The progressive scientific and technological evolution requires of teachers new ways to teach their students. In particular, the present work makes interesting connections between Mathematics and Informatics,extracting this union tools that stimulate curiosity, improve mathematical reasoning, making easy contents and concepts, and mainly allow the discovery of the path of resolution. In this way, we intend to show the true face of this teaching technique, by using Informatics, basing it and giving examples of how to implement activities that enable the student to develop mathematical concepts and processes, ability to solve problems, to reason, to communicate mathematically and as citizens. This research is based on bibliographical works that addresses issues related to teaching and learning of mathematical process.Some examples demonstrate that the mediation of computers brings new possibilities in solving problems. Concerning the last claim, it is expected that mathematics contribute to the formation of a citizen in its fullness, providing their integration as citizens in the workplace. / O enfoque principal deste trabalho é apresentar a resolução de problemas amparados por softwares como uma alternativa pedagógica para o ensino e a aprendizagem de conteúdos da Matemática. Para tanto, relaciona os objetivos dos PCN´s, a técnica descrita por Pólya em sua obra “How to solve it”, juntamente com possibilidade de uso de programas como: GeoGebra, Graphmatica, Planilhas Eletrônicas e Maxima. Sabe-se que muitos educadores matemáticos demonstram interesse por este assunto, porém, apesar de todo entusiasmo, poucos conseguem colocá-lo em prática no processo ensinoaprendizagem. A resolução de problemas ainda vigora como uma atividade complementar, sendo às vezes confundida com a resolução de exercícios, onde a resposta não é analisada e os valores encontrados não possuem significado para o aluno. A progressiva evolução científica e tecnológica demanda dos professores novas formas de ensinar seus alunos. Neste trabalho em especial, relacionam-se Matemática e Informática, extraindo desta união ferramentas capazes de estimular a curiosidade, melhorar o raciocínio matemático, tornar conteúdos e conceitos mais tangíveis e principalmente propiciar o gosto pela descoberta do caminho da resolução. Assim pretende-se demonstrar a verdadeira face desta técnica de ensino, utilizando a Informática, embasando-a e dando exemplos de como proceder para pôr em prática atividades que promovam nos alunos o desenvolvimento de conceitos, processos matemáticos e da capacidade de resolver problemas, de raciocinar, de comunicarem-se matematicamente e como cidadãos. A pesquisa realizada foi bibliográfica, baseada em obras que abordam questões inerentes ao processo ensino-aprendizagem da Matemática. Demonstra-se através de exemplos que a mediação de softwares trás novas possibilidades para resolução de problemas. Nesse aspecto espera-se que a Matemática como ciência contribua para a formação de um cidadão na sua plenitude, possibilitando de fato sua inserção, como cidadãos, no mundo do trabalho.

We investigate the famous Laguerre–Pólya class of entire functions and its subclass, the Laguerre–Pólya class of type I. The functions from these classes can be expressed in terms of the Hadamard Canonical Factorization (see Chapter 1, Definition 1.2 and 1.3). The prominent theorem by E. Laguerre and G. Pólya gives a complete description of the Laguerre–Pólya class and the Laguerre–Pólya class of type I, showing that these classes are the respective closures in the topology of uniform convergence on compact sets of the set of real polynomials having only real zeros (that is, the set of so-called hyperbolic polynomials) and the set of real polynomials having only real negative zeros. Both the Laguerre–Pólya class and the Laguerre–Pólya class of type I play an essential role in complex analysis. For the properties and characterizations of these classes, see, for example, [31] by A. Eremenko, [40] by I.I. Hirschman and D.V. Widder, [43] by S. Karlin, [57] by B.Ja. Levin, [66, Chapter 2] by N. Obreschkov, and [74] by G. Pólya and G. Szegö. In the thesis, we study entire functions with positive coefficients and with the monotonic sequence of their second quotients of Taylor coefficients. We find necessary and sufficient conditions under which such functions belong to the Laguerre–Pólya class (or the Laguerre–Pólya class of type I).:List of symbols Introduction 1 Background of research 1 1.1 The Laguerre–Pólya class .................... 1 1.2 The quotients of Taylor coefficients ............... 3 1.3 Hutchinson’s constant ...................... 4 1.4 Multiplier sequences ....................... 4 1.5 Apolar polynomials........................ 8 1.6 The partial theta function .................... 10 1.7 Decreasing second quotients ................... 13 1.8 Increasing second quotients ................... 14 2 A necessary condition for an entire function with the increasing second quotients of Taylor coefficients to belong to the Laguerre–Pólya class 15 2.1 Proof of Theorem 2.1....................... 16 2.2 The q-Kummer function ..................... 29 2.3 Proof of Theorem 2.10 ...................... 31 2.4 Proof of Theorem 2.11 ...................... 43 3 Closest to zero roots and the second quotients of Taylor coefficients of entire functions from the Laguerre–Pólya I class 49 3.1 Proof of Statement 3.1 ...................... 50 3.2 Proof of Theorem 3.2....................... 53 3.3 Proof of Theorem 3.4....................... 61 3.4 Proof of Theorem 3.6....................... 66 4 Entire functions from the Laguerre–Pólya I class having the increasing second quotients of Taylor coefficients 69 4.1 Proof of Theorem 4.1....................... 70 4.2 Proof of Theorem 4.3....................... 76 5 Number of real zeros of real entire functions with a non-decreasing sequence of the second quotients of Taylor coefficients 81 5.1 Proof of Theorem 5.1....................... 82 5.2 Proof of Corollary 5.2....................... 88 5.3 Proof of Theorem 5.4....................... 88 6 Further questions 95 Acknowledgements 97 Selbständigkeitserklärung 101 Curriculum Vitae 103 Bibliography 107

We present an algorithm for imputation of incomplete datasets based on Bayesian exchangeability through Pólya sampling. Each (donee) unit with a missing value is imputed multiple times by observed (real) values on units from a donor pool. The donor pools are constructed using auxiliary variables. Several features from kernel estimation are used to counteract unbalances that are due to sparse and bounded data. Three balancing features can be used with only one single continuous auxiliary variable, but an additional fourth feature need, multiple continuous auxiliary variables. They mainly contribute by reducing nonresponse bias. We examine how the donor pool size should be determined, that is the number of potential donors within the pool. External information is shown to be easily incorporated in the imputation algorithm. Our simulation studies show that with a study variable which can be seen as a function of one or two continuous auxiliaries plus residual noise, the method performs as well or almost as well as competing methods when the function is linear, but usually much better when the function is nonlinear. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: In press. Paper 3: Submitted. Paper 4: Submitted.</p>

Submitted by WILLIAN DIEGO OLIVEIRA null (willian@ibilce.unesp.br) on 2017-09-18T03:59:17Z No. of bitstreams: 1 Tese Final.pdf: 21063949 bytes, checksum: 766c3ca9aab9ca1a33dd27bf06043b1d (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-09-19T19:05:58Z (GMT) No. of bitstreams: 1 oliveira_wd_dr_sjrp.pdf: 21063949 bytes, checksum: 766c3ca9aab9ca1a33dd27bf06043b1d (MD5) / Made available in DSpace on 2017-09-19T19:05:58Z (GMT). No. of bitstreams: 1 oliveira_wd_dr_sjrp.pdf: 21063949 bytes, checksum: 766c3ca9aab9ca1a33dd27bf06043b1d (MD5) Previous issue date: 2017-05-11 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Estudamos tópicos relacionados a zeros de séries de Dirichlet e de funções inteiras. Boa parte da tese é voltada à localização de zeros de séries de Dirichlet via critérios de densidade. Estabelecemos o critério de Nyman-Beurling para uma ampla classe de séries de Dirichlet e o critério de Báez-Duarte para L-funções de Dirichlet em semi-planos R(s)>1/2, para p ∈ (1,2], bem como para polinômios de Dirichlet em qualquer semi-plano R(s)>r. Um análogo de uma cota inferior de Burnol relativa ao critério de Báez-Duarte foi estabelecido para polinômios de Dirichlet. Uma das ferramentas principais na prova deste último resultado é a solução de um problema extremo natural para polinômios de Dirichlet inspirado no resultado de Báez-Duarte. Provamos que os sinais dos coeficientes de Maclaurin de uma vasta subclasse de funções inteiras da classe de Laguerre-Pólya possuem um comportamento regular. / We study topics related to zeros of Dirichlet series and entire functions. A large part of the thesis is devoted to the location of zeros of Dirichlet series via density criteria. We establish the Nyman-Beruling criterion for a wide class of Dirichlet series and the Báez-Duarte criterion for Dirichlet L-functions in the semi-plane R(s)>1/p, for p ∈ (1,2], as well as for zeros of Dirichlet polynomials in any semi-plane R(s)>r. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to Báez-Duarte’s one is also established. A principal tool in the proof of the latter result is the solution of a natural extremal problem for Dirichlet polynomials inspired by Báez-Duarte’s result. We prove that the signs of the Maclaurin coefficients of a wide class of entire functions that belong to the Laguerre-Pólya class posses a regular behavior. / FAPESP: 2013/14881-9

Cette thèse porte sur l'analyse d'algorithmes stochastiques et leur application en Finance notamment et est composée de deux parties. Dans la première partie, nous présentons un résultat de convergence pour des algorithmes stochastiques où les innovations vérifient une hypothèse de moyennisation avec une certaine vitesse. Nous l'appliquons ensuite à différents types d'innovations (suites i.i.d., suites à discrépance faible, chaînes de Markov homogènes, fonctionnelles de processus \alpha-mélangeant) et nous l'illustrons à l'aide d'exemples motivés principalement par la Finance. Nous établissons ensuite un résultat de vitesse ''universelle'' de convergence dans le cadre d'innovations équiréparties dans [0,1]^q et nous confrontons nos résultats à ceux obtenus dans le cadre i.i.d.. La seconde partie est consacrée aux applications. Nous présentons d'abord un problème d'allocation optimale appliqué au cas d'un nouveau type de place de trading: les {\em dark pools}. Ces places proposent un prix d'achat (ou de vente) certain, mais n'assurent pas le volume délivré. Le but est alors d'exécuter le maximum de la quantité souhaitée sur ces places. Ceci mène à la construction d'un algorithme stochastique sous contraintes à l'aide du Lagrangien que nous étudions dans les cadres d'innovations i.i.d. et moyennisantes. Le chapitre suivant présente un algorithme d'optimisation pour trouver la meilleure distance de placement d'ordres limites: il s'agit de minimiser le coût d'exécution d'une quantité donnée. Ceci mène à la construction d'un algorithme stochastique sous contraintes avec projection. Pour assurer l'existence et l'unicité de l'équilibre, des critères suffisants sur certains paramètres du modèle sont obtenus à l'aide d'un principe de monotonie opposée pour les diffusions unidimensionnelles. Le chapitre suivant porte sur l'implicitation et la calibration de paramètres dans des modèles financiers. La première technique mène à un algorithme de recherche de zéro et la seconde à une méthode de gradient stochastique. Nous illustrons ces deux techniques par des exemples d'applications sur 3 modèles: le modèle de Black-Scholes, le modèle de Merton et le modèle pseudo-CEV. Enfin le dernier chapitre porte sur l'application des algorithmes stochastiques dans le cadre de modèles d'urnes aléatoires utilisés en essais cliniques. A l'aide des méthodes de l'EDO et de l'EDS, nous retrouvons les résultats de consistance (convergence p.s.) et de normalité asymptotique (TCL) de Bai et Hu mais sous des hypothèses plus faibles sur les matrices génératrices. Nous étudions aussi un modèle ''multi-bras'' pour lequel nous retrouvons le résultat de convergence p.s. et nous montrons un nouveau résultat de normalité asymptotique par simple application du TCL pour les algorithmes stochastiques.

In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach functionsetting. The thesis consists of three papers (A, B and C) and an introduction, which put these papers into a more general frame. This introduction has also independent interest since it shortly describe the dramatic more than 100 years of development of Hardy-type inequalities. It contains both well-known and very new ideas and results. In paper A we prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator, which is introduced in this paper (this operator generalizes the usualHardy kernel operator). These results generalize and unify several classical Hardy-type inequalities. In paper B we prove some new refined Hardy-type inequalities again in Banach function space settings. The used (super quadraticity) technique is also illustrated by making refinements of some generalized forms of the Jensen, Minkowski and Beckenbach-Dresher inequalities. These results both generalize and unify several results of this type. In paper C for the case 0&lt;p≤q&lt;∞ we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant. / In this thesis we derive various generalizations and refinements of some classical inequalities in different function spaces. We consider some of the most important inequalities namely the Hardy, Pólya-Knopp, Jensen, Minkowski and Beckenbach-Dresher inequalities. The main focus is put on the Hardy and their limit, Pólya-Knopp inequalities. Indeed, we derive such inequalities even in a general Banach function setting.  We prove and discuss some new Hardy-type inequalities in Banach function space settings. In particular, such a result is proved and applied for a new general Hardy operator. These results generalize and unify several classical Hardy-type inequalities. Next, we prove some new refined Hardy-type inequalities again in Banach function space settings. We used superquadraticity technique to prove refinements of some classical inequalities. Finally, for the case 0&lt;p≤q&lt;∞, we prove some new Pólya-Knopp inequalities in two and higher dimensions with good two-sided estimates of the sharp constants. By using this result and complementary ideas it is also proved a new multidimensional weighted Pólya-Knopp inequality with sharp constant.

Cette thèse étudie les urnes de Pólya à travers le prisme de la combinatoire analytique. Les urnes sont des modèles, conceptuellement très simples, de dynamique de croissance ou d'extinction dont les comportements limites sont extrêmement variés. Ces modèles sont largement étudiés par des approches probabilistes mais la compréhension précise des diverses lois limites reste une question ouverte. Les travaux de Flajolet et al. en 2005 ont illustré que pour ces questions, une approche par combinatoire analytique peut se révéler très fructueuse: l'étude des propriétés (nature, singularités) des séries génératrices associées aux urnes donne accès à des lois limites avec grande précision. Cette thèse s'inscrit dans la continuité de ces travaux et commence par identifier les séries des urnes de nature algébrique, grâce à un algorithme sophistiqué issu du calcul formel (Divination/Preuve automatique). Pour les classes d'urnes algébriques, nous menons des analyses, exacte et asymptotique, afin de connaître avec précision les comportements limites (structures des moments, vitesse de convergence, aspects limites locaux). Puis, l'étude d'urnes non algébriques est faite au travers d'exemples concrets portant sur la modélisation de réseaux sociaux, ainsi que sur la combinatoire des formules booléennes. Enfin, à travers des modèles d'urnes plus généraux (absence d'équilibre, présence d'aléa au sein des règles de substitution), nous montrons que l'approche symbolique de la combinatoire analytique est robuste. En particulier, une étude combinatoire générale des urnes sans condition d'équilibre est réalisée pour la première fois, unissant toute urne à une équation aux dérivées partielles.